# Functions of several variables

### Functions of several variables

#### Lessons

Notes:

2 Variable Functions
Functions with two variables are written in 2 forms:

$z = f(x,y)$
$F(x,y,z) = 0$

Domain & Range
The domain and range of a function $f(x,y)$ is as follows:

$D = \{ (x,y) \in \mathbb{R}^2 : f(x,y) \}$
$R = \{ z \in \mathbb{R} : f(x,y) = z \}$

Trace

A trace of a surface is a set of intersecting points which is created by a surface intersecting one of the following planes: $xy$-plane, $xz$-plane, $yz$-plane.

Trace of the $xy$-plane happens at $z=0$.
Trace of the $xz$-plane happens at $y=0$.
Trace of the $yz$-plane happens at $x=0$.

Level Curve
A level curve is a curve that is on the $xy$-plane with a specific value $z=z_0$.

• Introduction
Functions of Several Variables Overview:
a)
2 Variable Functions
• How to represent them
• What they look like Visually

b)
Domain & Range
• Domain $\to$ looks at $x$ and $y$
• Range $\to$ looks at $z$
• Examples of finding Domain & Range

c)
Traces
• Traces $\to$ intersection of surface and planes
• What it looks like visually
• Example of finding the Trace

d)
Level Curves
• Curves on the $xy$-plane with a specific $z$ value
• Examples of finding level curves

• 1.
Finding Domain & Range
Find the domain and range of:

$f(x,y) = \frac{1}{\sqrt{x^2 + y^2 - 4}}$

• 2.
Find the domain and range of:

$f(x,y) = \sqrt{x} \sqrt{x^2+y^2} e^{1-y^2}$

• 3.
Find the domain and range of:

$f(x,y) = ln(x^2 - 1)$

• 4.
Finding the Traces
Find and graph the trace of the $xz$ plane of

$f(x,y) = \sqrt{x}\sqrt{x+y^2}$

• 5.
Find and graph the trace of the $yz$ plane of

$5x + 2y + z = 3$

• 6.
Find the level curve of $f(x,y) = \sqrt{x^2 + y^2}$ at $z_0 = 1$ and $z_1 = 2$

• 7.
Find and graph the level curve of $2z + 4y^2 - x = 0$ at $z_0 = 0$.